In this paper, we developed a filtered version of the acyclic carrier theorem, which allowed us to construct interleavings between different geometric constructions. We have presented several results for Vietoris-Rips cover complexes, and we anticipate that the use of filtered carriers has broad potential as a technique to construct interleavings in situations that we have not yet considered. We have focused on algebraic interleavings, and many of these results could potentially be extended to homotopy interleavings [1]} given additional care when constructing carriers of cell complexes. Another interesting line of future investigation would be to use the algorithmic construction of maps from carriers
in data analysis. This could potentially be used, for instance, in constructing low dimensional embeddings of data that minimize the interleaving distance between a filtration on the higher-dimensional point cloud and the embedded point cloud.
